Hilbert

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11: 27.13 Functions

… ►Hilbert (1909) proves the existence of $g\left(k\right)$ for every $k$ but does not determine its corresponding numerical value. …

12: Bibliography O

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Numerical solution of Riemann-Hilbert problems: Painlevé II. Found. Comput. Math. 11 (2), pp. 153–179.

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External Links:

ISSN 1615-3375, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§32.17, §32.17.

Encodings:

BibTeX

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13: Bibliography D

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Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, Vol. 3, New York University Courant Institute of Mathematical Sciences, New York.

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External Links:

ISBN 0-9658703-2-4; 0-8218-2695-6, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt

Referenced by:

§18.38(ii), §18.38(ii).

Encodings:

BibTeX

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Linear operators. Part II. Wiley Classics Library, John Wiley & Sons, Inc., New York.

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Notes:

Spectral theory. Selfadjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1963 original, A Wiley-Interscience Publication

External Links:

ISBN 0-471-60847-5, MathReview Entry

Referenced by:

(1.18.68), §1.18(ix), §1.18(ix), §1.18(ix), §1.18(x).

Encodings:

BibTeX

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14: Bibliography H

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Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl $n^{\mathrm{ter}}$ Potenzen (Waringsches Problem). Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, pp. 17–36 (German).

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Notes:

Also in Math. Annalen (1909), v. 67, pp. 281–300.

External Links:

Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.13(iii).

Encodings:

BibTeX

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15: Bibliography W

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Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.

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External Links:

ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt

Referenced by:

§18.29.

Encodings:

BibTeX

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16: Bibliography S

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Multiparameter spectral theory in Hilbert space. Research Notes in Mathematics, Vol. 22, Pitman (Advanced Publishing Program), Boston, Mass.-London.

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External Links:

ISBN 0-8224-8414-5, MathReview (F. H. Szafraniec), ZentralBlatt

Referenced by:

Profile Brian D. Sleeman.

Encodings:

BibTeX

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Linear transformations in Hilbert space. American Mathematical Society Colloquium Publications, Vol. 15, American Mathematical Society, Providence, RI.

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Notes:

Reprint of the 1932 original

External Links:

ISBN 0-8218-1015-4, Document, Link, MathReview (P. R. Halmos)

Referenced by:

§1.18(x).

Encodings:

BibTeX

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17: 14.30 Spherical and Spheroidal Harmonics

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14.30.8_5 $${\mathrm{e}}^{t𝐚\cdot 𝐱}=\sqrt{4\pi }\sum _{n=0}^{\mathrm{\infty }}\sum _{m=-n}^n\frac{t^n{r}^n{\lambda }^m{Y}_{n,m}(\theta ,\varphi )}{\sqrt{(2n+1)(n+m)!(n-m)!}},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $n$: nonnegative integer and $m$: integer

Source:

Courant and Hilbert (1953, §VII.7.3)

Permalink:

http://dlmf.nist.gov/14.30.E8_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

Suggested 2019-09-26 by Anupam Garg

See also:

Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

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18: 18.2 General Orthogonal Polynomials

… ►A system $\{p_n(x)\}$ of OP’s satisfying (18.2.1) and (18.2.5) is complete if each $f(x)$ in the Hilbert space $L_w^2((a,b))$ can be approximated in Hilbert norm by finite sums ${\sum }_n{\lambda }_n{p}_n(x)$. …

19: Bibliography B

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Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 (1), pp. 185–266.

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External Links:

ISSN 0003-486X, Document, MathReview (Thomas Kriecherbauer), ZentralBlatt

Referenced by:

§18.32.

Encodings:

BibTeX

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20: Bibliography C

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Methods of mathematical physics. Vol. I. Interscience Publishers, Inc., New York, N.Y..

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External Links:

MathReview (J. B. Diaz), ZentralBlatt

Referenced by:

(14.30.8_5).

Encodings:

BibTeX

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